Time series copulas can also be used for discrete-valued data; see [1]} for
an early exploration of such models. [2]} do
so for longitudinal data using the extended likelihood at (REF ) and the
copula decomposition above, so that for \(\text{$y$}=(y_1,\ldots ,y_T)^\top \)  and \(\text{$u$}=(u_1,\ldots ,u_T)^\top \) ,
\(f_{Y,U}(\text{$y$},\text{$u$}) &= &\prod _{t=1}^T\left\lbrace {1}(a_t \le u_t < b_t)\right\rbrace c_{1:T}(\text{$u$}) \nonumber \\&= &\prod _{t=2}^T\left\lbrace  f_{U_{t|1:t-1}}(u_{t}|u_1,\ldots ,u_{t-1}) {1}(a_t \le u_t < b_t)\right\rbrace {1}(a_1 \le u_1 < b_1)\,,\) 
